What's the precise meaning of imaginary number?

Question

The same to the title,what's the precise meaning of imaginary number? And on the other hand,how can the imaginary number be reflected in Physics?

Answer 1

I'm unsure what you mean by the "precise" meaning of an imaginary number, but to me it seems best to talk about the prototypical example of an imaginary number, the imaginary unit $i$, and then work from there.

You can think of the imaginary unit as being a solution of the equation $X^2+1=0$ i.e. you define $i$ to be a number such that $i^2=-1$ (note that I cannot say "the" number since it is not unique, as $-i$ also has this property). To me, this is the precise meaning of the imaginary unit. One then defines the complex numbers $\mathbb{C}$ in terms of this imaginary unit as

$$\mathbb{C}=\{a+bi \; |\; a,b\in \mathbb{R}\}.$$

More formally, the complex numbers are obtained by adjoining a root of $X^2+1$ to $\mathbb{R}$ by taking the quotient of $\mathbb{R}[X]$ by the maximal ideal $(X^2+1)$. Let $C= \mathbb{R}[X]/(X^2+1)$. Note that $\{1,X\}$ is a basis for $C$ and $X$ has the property that $X^2=-1$. From this it is seen that the map $\varphi:C \rightarrow \mathbb{C}$ defined by $\varphi(a+bX)=a+bi$ is an isomorphism of fields.

Another way of viewing the complex numbers is as follows:

The set $R$ of matrices of the form

$$\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix},$$

where $a \in \mathbb{R}$, behaves exactly like the real numbers with respect to the operations of matrix addition and multiplication i.e. they are isomorphic as fields ($R \cong \mathbb{R}$ ). Consider the matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Notice that this matrix has the property that

$$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}=\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}.$$

Setting $i=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and using our identification above, we see that $i$ is a solution of the equation $X^2+1=0$. So with this one could reasonably believe that the precise meaning of the imaginary unit $i$ is $i=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. One then notices that the set of matrices $C$ of the form

$$\begin{pmatrix} a & -b \\ b & a \end{pmatrix},$$

where $a,b \in \mathbb{R}$ behave precisely like the complex number $\mathbb{C}$ i.e. they are isomorphic as fields ($C\cong \mathbb{C}$), and takes this to be the precise meaning of the complex numbers.

Additionally, one could formally develop the complex numbers by defining $C$ to be the set of pairs $(a,b) \in \mathbb{R}^2$ such that additions and multiplication are defined by

$$(a,b)+(c,d)=(a+c,b+d)$$ and $$(a,b)(c,d)=(ac-bd,ad+bc).$$

First note that subset $D$ of $C$ consisting of elements of the form $(a,0)$ behaves just like elements of $\mathbb{R}$, so, in particular, $(1,0)$ is the multiplicative identity of $C$. One then notices that the element $(0,1) \in C$ has the property that $(0,1)(0,1)=(-1,0)$ and so it is a solutions to the equation $X^2+1=0$.

Answer 2

This is a mathematician's view.

At heart, anything involving symmetry involves group theory. We like to represent our groups as matrices. One of the most powerful ways to study matrices is to study the eigenvalues and eigenvectors of the matrices.

The eigenvalues of real matrices can still be complex, however. So if we are studying matrices of symmetries in real vector spaces, we often end up "breaking up" the elements of that space into components that are complex, which has the paradoxical affect of simplifying the study of the underlying matrix.

The further you get into physics, the more you become interested in various types of symmetries, and there is really no way to study those symmetries without looking at complex numbers. After a while, you start to think of these complex components as parts of the underlying universe that we just don't see because they "cancel out."

Answer 3

Studying some LRC circuits might be a good way to get an idea of how complex numbers can be used in physics. As opposed to Quantum Mechanics, LRC circuits can be studied by the average first or second year undergraduate. Yale has a couple of pretty decent lectures on them.

Answer 4

The precise meaning of a complex number is that it is a particular complex number. It is what it is, no more, no less.

When one uses numbers (or other sorts of objects) to quantify things, one shouldn't confuse the number with the thing being quantified. For example, the picture

 * * * * *

is not the number "5". It is simply a collection of asterisks. By using "5" to quantify the number of asterisks, we can better understand and work with the picture. Sometimes we can use such a picture to help us understand and work with "5". But "5" is "5" and the picture is the picture, and you shouldn't confuse the two.

Complex numbers are often useful for quantifying a variety of things, such as various properties of waves, the impedance of a circuit, or position on a planar surface. They often can be used to simplify formulas and calculations. These are all uses of a complex number.

But what is a number like "$1 + 3 \mathbf{i}$"? "$1 + 3 \mathbf{i}$" is simply "$1 + 3 \mathbf{i}$", just like "5" is just "5".

Answer 5

In the first place, an imaginary or complex number is an element of the system ${\mathbb C}$ of complex numbers and as such is an object of our imagination.

In the last five hundred years or so mathematicians have come to the conclusion that the introduction of the system ${\mathbb C}$ into the body of everyday mathematics brings overwhelming benefits to the understanding of polynomials and various higher elementary functions (in particular the exponential and trigonometric functions), differential equations, etcetera.

Answer 6

Physically, imaginary number quantities do not exist. However, it is often very convenient to use them for intermediate steps in extensive calculations, for example you can take the complex exponential $e^{i\omega t}$ to describe harmonic waves in linear systems where you can superimpose different solutions. That means if both a complex valued function and its complex conjugate are valid solutions of a (set of) linear (partial or ordinary) differential equation(s), then their real and imaginary part alone (being $\propto f\pm\bar f$) are also solutions, but since trigonometric identities are a lot messier than exponential ones, using complex numbers is very useful.

There is one possible interpretation of complex numbers though, if you do not take it too literally: Time. Relativistic Minkowsky space-time has a metric $ds^2 = dx^2 + dy^2 + dz^2 - \frac1{c^2}dt^2$. You could turn this into an Euclidean metric by considering time as an imaginary quantity^*, e.g. $\beta := \frac i{\hbar} t$. The funny thing is, this Wick rotation (that's why I use Planck's constant $\hbar$ here) turns many quantum mechanical equations into thermodynamic ones by replacing imaginary time with the inverse temperature $\beta$ and vice versa, yielding some interesting interpretations of both theories...

^*_{You can also replace the spacial coordinates by the three imaginary units $i,j,k$ of quaternions...}

Answer 7

Let me address your first question about "what's the precise meaning of imaginary number?". First though, in my opinion, we have to be careful when we start to give meaning to mathematical objects. When we talk about "meaning" then the answer, IMO, becomes subjective and so to ask for the precise meaning doesn't really make sense. You are trying to relate it to something else - something that is different.

What we can do is to give the precise definition of what an imaginary number is. And we do that when we define what the complex numbers are. When we have done that, we can then talk about varies ways of thinking about this definition to get an intuition about what the complex numbers are. Or we can find that certain things in say physics have a structure to them that makes us think of complex numbers. But when we do that we (IMO) often leave the definition and so we leave the preciseness.

Answer 8

Maybe we can think of our numbers as ways to navigate a particular space. So natural numbers, integers, rational numbers, and real numbers allow us to reach various points of a line. Adding complex numbers allows us to rotate out of the line, and reach any point on the plane. A particular complex number is a point in the plane. Add another complex number to that and you can get to another. Multiply it with a complex number, and you can "stretch" and "twist" the vector to get to another point again.. Later, Quaternions, Octonions, etc allow you to get out of the plane.